Non-triviality of the A-polynomial for knots in S³

Abstract

The A-polynomial of a knot in S³ defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL₂C. Here, we show that a non-trivial knot in S³ has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU₂-representations of Dehn surgeries on knots in S³. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot.

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